Resistance of superconducting nanowires connected to normal-metal leads
G. R. Boogaard, A. H. Verbruggen, W. Belzig,*and T. M. Klapwijk
Kavli Institute of Nanoscience, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
(Received 20 January 2004; revised manuscript received 20 April 2004; published 18 June 2004)
We study experimentally the low temperature resistance of superconducting nanowires connected to normal metal reservoirs. We find that a substantial fraction of the nanowires is resistive, down to the lowest tempera-ture measured, indicative of an intrinsic boundary resistance due to the Andreev-conversion of normal current to supercurrent. The results are successfully analyzed in terms of the kinetic equations for diffusive superconductors.
DOI: 10.1103/PhysRevB.69.220503 PACS number(s): 74.78.Na, 74.45.⫹c, 85.25.Pb
Superconducting nanowires are resistive down to very low temperatures due to dynamical changes of the macro-scopic phase (phase-slip). Thermally activated phase-slip (TAP) becomes more likely for reduced cross-sectional di-mensions because the free-energy barrier scales with the area of the wire. Upon approaching T→0, phase slip due to ther-mal activation disappears and resistivity persists only by macroscopic quantum tunneling through the free-energy bar-rier. These processes have recently been studied in sus-pended carbon nanotubes coated with a thin layer of a super-conducting molybdenum-germanium 共MoGe兲 alloy,1,2 and
receive increased theoretical attention.3,4 In a separate
experiment5 ropes of single-walled carbon nanotubes show
signs of superconductivity, which should be strongly influ-enced by phase-slips as well.
A second potential cause of low temperature resistance in superconducting wires is the penetration of a static electric field at normal-metal–superconductor interfaces. It reflects the conversion of current carried by normal electrons into one carried by Cooper-pairs via Andreev-reflection. It has been studied extensively close to the critical temperature, where it is related to quasiparticle charge imbalance.6,7
Al-though hardly experimentally studied, at very low tempera-tures a similar resistive contribution is expected to be present, reflecting a length of the order of the coherence length. Since the coherence length is a sizable portion of the resistive length of the nanowires it may contribute signifi-cantly to the measured two-point resistance. Here we report experimental results on the resistance of narrow supercon-ducting wires connected to normal metal leads (NSN, for short). We find a strong contribution to the resistance due to the conversion processes, which is analyzed and understood in terms of the nonequilibrium theory for dirty superconduct-ors.
We chose to study samples (Fig. 1) made of supercon-ducting (S) aluminium 共Al兲 because of its long coherence length. Our main interest is in the two-point resistance of the S-wire. Hence, we have chosen to work with thick and wide normal 共N兲 contacts with a negligible contribution to the normal state resistance. To minimize interface resistances due to electronic mismatch of both materials, we have cho-sen to work for N with bilayers of aluminium covered with thick normal metal共Cu兲. In such a geometry the
supercon-ducting aluminium wire is directly connected to normal alu-minium.
The S-wire of 100 nm thick Al is made by evaporating 99.999% purity Al at a rate of ⬃1 nm/s in a vacuum of 1 ⫻10−8mbar during evaporation. Films made in the same
way, have a residual resistance ratio, RRR= R300 K/ R4.2 Kof 7.5 indicative of the level of impurity scattering. Taking the phonon resistivity of ph共Al兲=2.7⍀ cm, the impurity
re-sistivity is 0= 0.4⍀ cm. Using 0= N共0兲e2D and
renor-malized free-electron parameters: N共0兲=2.2⫻1047J−1m−3
andvF= 1.3⫻106 m / s(Ref. 8), we find the elastic mean free
pathᐉ of 100 nm, presumably limited by the thickness. The superconducting transition temperature of the 100 nm film is 1.26 K, the usually enhanced value for aluminium thin films. The sample is made in one evaporation run using shadow evaporation. A 300 nm PMMA/ 500 nm PMMA-MAA lift-off mask defines the width and the length of the S-wire. The width is approximately 250 nm and the length is varied from 1m to 4 m. The wire and the aluminium of the normal reservoirs is deposited in one run, both 100 nm thick. The reservoirs are made normal by a second evaporation of 470 nm copper共Cu兲 on top. The Cu is evaporated at a rate of ⬃2.5 nm/s at a pressure of 1⫻10−7mbar. The
material-parameters are 0= 0.4⍀ cm, ᐉ=165 nm, D = 66
FIG. 1. SEM picture of a device(slightly misaligned), showing the coverage of the thin aluminium film with the thick Cu layer
(except for one the measured devices are carefully lined up). The
inset shows a schematic picture of an ideal device. PHYSICAL REVIEW B 69, 220503(R) (2004)
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⫻10−3m2/ s using N共0兲=1.5⫻1047 J−1m−3, and
vF= 1.2
⫻106m / s. The nonsuperconducting properties of the final
reservoirs are confirmed by measuring the resistance of a 100 nm Al/ 470 nm Cu bilayer down to the lowest tempera-ture measured: 600 mK. No sign of superconductivity is ob-served. This is in agreement with the analysis of Tc for a
bilayer using the Usadel equations.9Only for a transparency
of the Al/ Cu interface lower than 0.1 the Tc would reach
values above 600 mK. 0.1 is an unrealistically low transpar-ency for an in vacuum prepared Al/ Cu interface.
The two-point resistance is probed with a current of 1A (linear regime) in a3He system down to 600 mK. The
volt-age over the wire is measured with a lock-in amplifier at 133 Hz.
We find that the normal state resistance of nominally iden-tical wires scatters substantially(10% or more), presumably due to grain-sizes compared to wire-widths. To allow a quan-titative evaluation we have selected a set of wires with iden-tical values of RRR= 3.2± 0.1. Figure 2 shows the R – T mea-surements for this particular set of wires. Samples with different RRR values show qualitatively identical behavior. All wires show a finite remaining resistance as most striking result.10
Obviously, despite the differences in length, the resis-tances at low temperatures have identical values and follow the same trace. It indicates that the origin of this remaining resistance is likely due to the region in the S-wire next to the interface with the normal reservoir. The resistance at 600 mK is equal to a normal segment of the superconductor with a length of about 200 nm.
Figure 2(inset) also shows that the critical temperature of the wire decreases linearly with increasing the inverse square of the wire length.
The critical temperature of the wire should follow a straightforward Ginzburg-Landau analysis. ForGL共T兲艋L
the bridge should become superconducting. This leads to an ⬃1/L2onset-temperature according to Tc Tc0 = 1 − 2.2 បD ⌬共0兲 1 L2. 共1兲
For L→⬁ the normal contact can no longer depress Tcand
we find the intrinsic critical temperature. For the studied wires it is found to be Tc0= 1.26 K, identical to the
indepen-dently determined values of the 100 nm Al film.
From the normal state resistance we infer an impurity resistance of 0= 1.1⍀ cm, in accordance with the RRR = 3.2. It is significantly higher than the0 of the 100 nm Al
film, leading to a lower diffusivity and elastic mean free path: D = 160 cm2/ s, andᐉ=37 nm, respectively. The
result-ing coherence length=
冑
បD/2kBTc= 124 nm.Finally, we estimate a resistance contribution of 11 m⍀ due to the spreading resistance in the normal reservoirs at low temperatures, considerably less than the value we mea-sure in Fig. 2.
Theoretically, since the studied nanowires show diffusive transport, the Usadel equations should apply to the system.11
It is most convenient to calculate the normal current for a given applied voltage difference, assuming linear response. Schmid and Schön12 have shown that within this limit the
normal current can be described with a variation,␦f共E,x兲, in the electronic distribution function f共E,x兲:
I =A e
冕
−⬁+⬁
M共E,0兲␦f共E,0兲
x dE 共2兲
with x the coordinate along the length of the wire.␦f共E,x兲 is determined from a Boltzmann-equation, which includes the conversion of electrons into Cooper-pairs, but ignores inelas-tic electron-phonon scattering(only relevant close to Tc):
បD x
冉
M␦f
x
冊
− 2⌬N2␦f = 0. 共3兲 The position-dependent spectral conductivity M共E,x兲 consist of two parts N1共E,x兲=Re共G兲 and N2共E,x兲=Re共F兲, with G the normal and F the anomalous Greens function:M共E,x兲 = 共N1共E,x兲兲2+共N2共E,x兲兲2, 共4兲
N1共E,x兲 is comparable to the standard BCS density of states.
The applied voltage V, is taken into account via the boundary conditions for Eq.(3),
␦f共E;0,L兲 = ±eV/2 4kBT cosh2共E/2kBT兲
. 共5兲
Hence, the normal metal leads are taken as equilibrium res-ervoirs. The strength of the pairing interaction ⌬共x兲 in Eq. (3) is determined by solving the Usadel equation in the Mat-subara representation: 1 2បD 2 x2= −⌬共x兲cos+nsin, n=共2n + 1兲kBT, n = 0,1, . . . 共6兲
in conjunction with the self-consistency equation
FIG. 2. (Color online) Measured R–T curves for four different bridge lengths. The intrinsic Tc0is indicated by the vertical dashed
line. The inset shows the measured critical temperature of the wire versus 1 / L2, which is used to determine Tc0by letting L→⬁.
BOOGAARD et al. PHYSICAL REVIEW B 69, 220503(R) (2004)
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⌬ ln
冉
Tc T冊
= 2kBT兺
n=0 ⬁冉
⌬ n − sin共兲冊
. 共7兲 The so-called proximity angle, , parametrizes the normal Green’s function G = cos and the anomalous Green’s func-tion F = sin. The spectral functions are calculated by again solving the Usadel equation but now for→−iE. The large normal metal reservoirs impose the boundary condition共x = 0 , L兲=0Our main interest is the question how the current conver-sion process contributes to the resistance. First of all, the presence of decaying normal electron states suppresses the gap in the density of states.
In Fig. 3, the density-of-states N1共E兲 is given for several positions along the wire of L = 2m, D = 160 cm2/ s, ⌬
0
= 192eV, and T / Tc= 0.4. Clearly, moving away form the
normal leads the density of states resembles more and more the well-known BCS density of states. Note however, that a finite subgap value remains in the middle共x=1m兲 even for very long wires. This is an intrinsic result for any NSN sys-tem and it is not due to current-flow, since this result is calculated in thermal equilibrium. (The back-action of the current-flow on the spectral properties can be neglected in the linear response regime.)
In Fig. 4, we show the results of a calculation of the voltage as a function of position along the wire for two dif-ferent temperatures: t = 0.4, and t = 0.9 with D = 160 cm2/ s, and⌬0= 192eV. At the temperature close to the transition
temperature, the electric field penetrates the sample com-pletely and the resistance is close to the normal state value. At low temperatures, the electric field still penetrates the superconductor over a finite length, leaving a middle piece with hardly any voltage drop.13The penetration length is of
the order of the coherence length. The inset shows the posi-tion dependent normal currents and supercurrents illustrating the current conversion processes.
In Fig. 5, a comparison is made between the calculated resistance as a function of temperature and the measurement for a L = 2m wire. The calculation is done with parameters D = 160 cm2/ s, as determined from the impurity resistivity,
and⌬共0兲=1.764kBTc= 192eV with Tc= 1.26 K determined from the length dependence. These parameters have been determined independently. Without any fitting parameter, the agreement between the model (dots) and the experiment (data points: open symbols) is encouragingly good. Only at the lower temperatures the observed resistance is slightly less than the theoretically predicted values.
FIG. 3. The calculated density-of-states N1at various distances
from the reservoirs 共x=100,200,300,400,500,1000 nm兲 for a 2m long wire (t=0.4, D=160 cm2/ s, and ⌬0= 192eV). Note
the exponentially small but finite subgap density-of-states in the middle(at x=1m; see inset).
FIG. 4. The voltage in the superconducting wire as a function of position for two different temperatures(t=0.4 and 0.9). At t=0.9 the wire behaves as a normal metal and for t = 0.4 the voltage is clearly present to a depth (wire length 2m with D = 160 cm2/ s and⌬0= 192eV). The inset shows the position
de-pendent normal currents and supercurrents.
FIG. 5. (Color online) The measured R–T curve for the 2m bridge together with the model calculation using the boundary con-dition 共x=0,L兲=0 (dots), and using the boundary condition
x共x=0,L兲=共x=0,L兲/a (triangles) with a=75 nm. The value for
T = 0 is 0.3255 for the hard boundary conditions(dots), and 0.2609
for the soft boundary conditions.
RESISTANCE OF SUPERCONDUCTING NANOWIRES… PHYSICAL REVIEW B 69, 220503(R) (2004)
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Apparently, the model is overestimating the remaining re-sistance below T = 0.8 K. Since there is little freedom left, we have very few options to remedy this discrepancy. The most likely option is that the rigid boundary conditions imposed at the interfaces should be relaxed. There is a finite possibility for superconducting correlations to extend into the normal metal reservoirs, which would mean that the boundary con-dition共x=0,L兲=0 is too rigid. Since the correlations will extend into a three-dimensional volume we assume that us-ing the boundary conditions x共x=0,L兲=共x=0,L兲/a, i.e.,
a decay over a fixed characteristic length a, is a realistic assumption. It assumes a geometric dilution of the correla-tions. The result is shown in Fig. 5 by filled triangles. The best agreement between measurement and calculation model is obtained for a = 75 nm. This value appears reasonable for a decay length since it is comparable to the dimensions (100 nm⫻250 nm) of the wire which emits into the reser-voirs. The shortest bridge shown in Fig. 2 is found to have a further reduction in resistance, which we attribute to a small misalignment as shown in Fig. 1. Note however that the model predicts that NSN devices will continue to be resistive down to T = 0 K. For the rigid boundary conditions we find R / R4.2K= 0.3255 and for the soft boundary conditions we
find R / R4.2 K= 0.2609.
An early indication of the low temperature boundary re-ported here is given by Harding et al.14 They studied the
resistance of thick sandwiches of Pb-Cu-Pb. By subtracting the contribution to the resistance of the Cu-layers they iden-tified a remaining boundary resistance which depended on
the mean free path in the Pb layers. In later work by Hsiang and Clarke7 such a resistance appeared to be unobservable,
in contrast however to more recent work by Gu et al.15 The
geometry of our sample, with a negligible contribution to the resistance from the reservoirs, allows us to measure only the resistance due to the conversion processes in the supercon-ductor. In response to the work of Harding et al. Krähenbühl and Watts-Tobin16 derived an analytical expression for the
effective length of the boundary resistance: x0=
冑
160lf共T兲
withᐉ the mean free path for elastic scattering,0 the BCS
coherence length, and f a function of temperature. For T = 0 K, the function f is of order 1. In contrast to our model Ref. 16 allows for one-dimensional diffusion in N, rather than treating N as an equilibrium reservoir.
In conclusion, we have shown that the resistance of a superconducting nanowire connected to normal leads has a finite dc resistance down to very low temperatures. The mi-croscopic theory describes the data very well and provide a detailed image of the conversion from normal current into supercurrent, over a few coherence lengths. The results em-phasize the important role of the length of the wires in rela-tion to the nature of the contacts.3 It also explains results
obtained with diffusion-cooled hot-electron bolometers in which normal leads are used to provide rapid out-diffusion of hot electrons from a superconducting wire.10
We thank the Stichting voor Fundamenteel Onderzoek der Materie(FOM) for financial support. We thank R. S. Keizer and A. A. Golubov for stimulating and clarifying discussions.
*Current address: Department of Physics and Astronomy, Univer-sity of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland.
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